By definition \(L_\infty (n_\theta )\)  is a von Neuman algebra of bounded operators on \(L_2(n)\) . Thus, a closed densely defined operator on \(L_2(n)\)  is \(L_\infty (n_\theta )\) -affiliated when it commutes with the commutant of \(L_\infty (n_\theta )\)  in \({L}(L_2(n))\) . Furthermore, as \(\tau \)  is a finite faithful positive trace on \(L_\infty (\theta ^n)\)  every such operator is \(\tau \) -measurable in the sense of [1]}, [2]}. Therefore, these operators form a \(*\) -algebra, where the sum and product of such operators are meant as the closures of their usual sum and product in the sense of unbounded operators (see [2]}).
